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Questions tagged [stochastic-processes]

For questions about stochastic processes, for example random walks and Brownian motion.

A stochastic process is a collection of random variables representing the evolution of a system of random variables over time. A typical example is a .

A stochastic, or random, process describes the correlation or evolution of random events. It is used to model stock market fluctuations and electronic/audio-visual/biological signals. Among the most well-known stochastic processes are random walks and Brownian motion.

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Understanding stochastic processes.

I'm working through my statistics course which includes some material on stochastic processes. Unfortunately the way it's explained is not giving confidence in my understanding so I would appreciate it if someone could help explain. From my course…
thetime
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If $W(t), t \ge 0$ Wiener process, show $ W(\lambda t) $ Wiener process, $\lambda \gt 0$

As title says. Intuitively it seems clear it should be true from the definition of Wiener process since $\lambda t \ge 0$, i.e it should be the "same" process shifted $\lambda$ in time. But it doesn't feel a rigorous enough statement.
apelsimon
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How to calculate the volatility of a compensated poisson process?

Poisson process $N(t)$ with density $\lambda$, could generate a compensated Poisson Process $$M(t) = N(t) - \lambda t,$$ $M(t)$ is a martingale with mean of $0$. Now, how could I calculate the volatility of this compensated Poisson process…
athos
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Limit of Wiener processes

Let $W_t$ be Wiener process. I am trying to evaluate the following limit $$\lim\limits_{n \to \infty}~{\sum\limits_{i=1}^{n}W_{\frac{i-1}{n}+\frac{1}{2n}}\left( W_{\frac{i}{n}} - W_{\frac{i-1}{n}} \right)}$$ I've expanded parethesis and got $$…
ivust
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What is the probability of no events in a Markov-modulated Poisson process?

Suppose I have a two-state continuous-time Markov chain $M$ with rate matrix $Q$. $$ Q = \begin{bmatrix} -q_{01} & q_{01} \\ q_{10} & -q_{10} \end{bmatrix} $$ Now consider a Poisson process $P$ whose rate is $\lambda_0$ when $M$ is in state 0, and…
rmccloskey
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Smith's Key Renewal Theorem for Renewal Function

Consider a renewal process $(N_t)_{t \geq 0}$ and its renewal function $M(t):=\mathbb{E}[N_t]$ with interarrival distribution function $F$. One can show that $M$ satisfies the $(F,F)$-renewal equation, i.e. satisifies $$M(t)=F(t) + M* F(t) = F(t)+…
user136457
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Sufficient conditions for a stochastic process to be continuous?

Let $(X_t)_{t\ge 0}$ be a stochastic process, such that $X_t-X_s \xrightarrow{w} \delta_0$ as $t\to s$. Is that a sufficient condition for $(X_t)_{t\ge 0}$ to be continuous? If not, can you provide what conditions should be needed. Also, can you…
user2808118
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Aggregate arrivals from a Poisson Process

The inter-arrival time of a Poisson Process, $t$, conforms to the exponential distribution, so the probability density function for $t$ is $f(t)=λe^{−λt},~t>0$. ($λ$ is the arrival rate of the Poisson Process.) Next we aggregate the requests…
Bloodmoon
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Represent stochastic process as conditional expectation

I try to reduce my problem to the following question, which is stated rather sloppy (without possibly necessary additional conditions). Let $Y_t$ be a real stochastic process for $t \in [0, T]$ and $\mathscr{F}_t$ some filtration. Does there exist a…
yada
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Probability: $p\{X_t\in A\mid \min_{0\leq u\leq t}B_u>a\}=p\{X_t\in A,\min_{0\leq u\leq h} B_u>a\mid \min_{h\leq u\leq t}B_u>a\}$ always work?

Let $(X_t)$ and $(B_t)$ two stochastic processes and $0\leq h\leq t$. Do we always have $$p\left\{X_t\in A\mid \min_{0\leq u\leq t}B_u>a\right\}=p\left\{X_t\in A,\min_{0\leq u\leq h} B_u>a\mid \min_{h\leq u\leq t}B_u>a\right\}\ \ \ ?$$ I have that…
idm
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Finding the transition density for $(W_t, B_{1,t}, B_{2,t})$, where $\mathrm{d} W_t = \frac{B_t}{| B_t |} \cdot \mathrm{d} B_t$

Let $B_t$ be 2-dimensional standard Wiener process. Define $W_t$ as $$ W_0 = 0, \quad W_t = \int_0^t \frac{B_s \cdot \mathrm{d}B_s}{\sqrt{B_s \cdot B_s}} $$ It is well known that $W_t$ is a standard 1-dimensional Wiener process. I convinced…
Sasha
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Path of diffusion process with discontinuous drift

Let $(B_t)$ be a standard Brownian motion on some probability space and let $X_t$ be the process defined by the SDE $dX_t = \mu_t dt + dB_t$, where $\mu_t$ is adapted, deterministic, and only takes the values $0$ or $1$. (For example, $\mu_t = 0$…
Victor
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Stationary Probabilities: Periodic Case: motivation 2nd attempt.

For DTMC with $S=\{1,2\}$ and transition probabilities $$P = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$$ How do we see that $(P_{00})^{(n)} = 1$ if $n$ is even or $0$ if $n$ is odd ?? (Where $P_{ij}$ is a one-step transition probability). I'd…
Max
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Impossible stochastic process

I am trying to prove that a stochastic process with the following properties cannot exist. Let $\{X_t: 0 \leq t \leq 1 \}$ be a stochastic process such that i) $X_s$ and $X_t$ are independent whenever $s\neq t$; ii) Each $X_t$ has the same…
Calculon
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Distribution of the increments of a Compound Poisson process

Let $X_t$ be a compound Poisson process defined as $X_t = \sum_{i=1}^{N_t} D_i$, where $D_i$ are i.i.d. and $D_i \sim Exp(\mu)$ and $N_t$ is a Poisson process with parameter $\lambda$. As usual the Poisson process is independent from $D_i$. Let…
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